# Non-linear system

Nonlinear systems ( NL systems ) are systems theory systems whose output signal is not always proportional to the input signal (system stimulus). They can be much more complex than linear systems .

## general basics

For non-linear systems, in contrast to linear systems, the superposition principle does not apply. This means that one cannot infer an unknown system response for a given system stimulus from several known system stimulus-system response pairs. Furthermore, one differentiates the non-linearity of a system into static , dynamic , single-valued and multi-valued non-linearity. Since there is no closed mathematical theory for nonlinear systems, there is also no general method for analyzing unknown nonlinear systems.

In general, a mathematical model of a non-linear system with an internal state , external influences and observations can be represented as ${\ displaystyle x (t)}$${\ displaystyle u (t)}$${\ displaystyle y (t)}$

{\ displaystyle {\ begin {aligned} {\ dot {x}} (t) & = A {\ bigl (} t, x (t), u (t) {\ bigr)} \\ y (t) & = C {\ bigl (} t, x (t), u (t) {\ bigr)}, \ end {aligned}}}

where and are the non-linear functions that describe the system. ${\ displaystyle A}$${\ displaystyle C}$

## Static nonlinear systems

Illustration of a linear (left diagram) versus a non-linear (right diagram) characteristic . The dashed diagonal illustrates the linear or non-linear transformation , the black curve is the input signal, the blue curve is the output signal.

Static non-linear systems are understood to be those that react to a system stimulus without a time delay. For example, the diode is generally viewed as a static component (with the exception of fast switching operations). Their voltage-current characteristic follows an exponential function; in various applications it is idealized and treated as piece-wise linear, but remains non-linear in the system-theoretical sense. Static systems can be described by a static characteristic curve as shown in the figures.

Characteristic curve of a field effect transistor
Above (for example at> 3 mA) almost linear: A sinusoidal course of a change Δ U GS generates a change Δ I D without a visible deviation from the sine curve.
Bottom (for example at <3 mA) non-linear: A sinusoidal course of a change Δ U GS generates a change Δ I D with a recognizable non-sinusoidal course.

## Dynamic nonlinear systems

Dynamic non-linear systems are those that also have storage elements and thus a “memory”. As a result, the system response is not determined by the instantaneous value of the system stimulus alone. It also depends on the previous history, i.e. on the strength of the previous arousal.

## Characterization with regard to the frequency behavior

When linear systems are excited with a sinusoidal signal, a sinusoidal signal of the same frequency is obtained at the output, but with a changed phase position and amplitude. Nonlinear systems generally do not have this property. Non-linear systems can have frequency components at their system output that are not contained in the input signal ( distortion ).

Examples from electrical engineering are:

• If a non-linear amplifier is fed with a single sinusoidal voltage as input voltage, it generates harmonics at the output in addition to a sinusoidal voltage . Their proportions increase with increasing overdrive .
• If the amplifier is fed with a superposition of two or more sinusoidal voltages of different frequencies, intermodulation also occurs, and combination frequencies arise.
• If several modulated AC voltages are to be amplified at the same time, cross modulation can occur. Then one alternating voltage partially takes over the modulation of the other ( Luxembourg effect ).

## literature

• Mathukumalli Vidyasagar: Nonlinear systems analysis SIAMm Philadelphia 2008, ISBN 978-0-89871-526-2 .
• Muthuswamy Lakshmanan, et al .: Nonlinear dynamics - integrability, chaos, and patterns. Springer, Berlin 2003, ISBN 3-540-43908-0 .

## Individual evidence

1. Holk Cruse: Biological Cybernetics. Verlag Chemie GmbH, Weinheim 1981, ISBN 3-527-25911-2 .
2. Dezsö Varjú: Systems Theory. Springer-Verlag, Berlin / Heidelberg 1977, ISBN 3-540-08086-4 .